BPS: Difference between revisions
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== Tuning spectrum == | |||
{| class="wikitable center-all left-4" | |||
! Edt<br>Generator | |||
! [[Eigenmonzo|Eigenmonzo<br>(Unchanged-interval)]] | |||
! Generator<br>(¢) | |||
! Comments | |||
|- | |||
| [[22edt|5\22]] | |||
| | |||
| 432.263 | |||
| | |||
|- | |||
| | |||
| 7/3 | |||
| 435.084 | |||
| 0-comma | |||
|- | |||
| [[61edt|14\61]] | |||
| | |||
| 436.514 | |||
| | |||
|- | |||
| | |||
| 3645/2401 | |||
| 437.449 | |||
| 1/6-comma | |||
|- | |||
| [[100edt|23\100]] | |||
| | |||
| 437.450 | |||
| | |||
|- | |||
| | |||
| [[49/45]] | |||
| 438.632 | |||
| 1/4-comma | |||
|- | |||
| [[13edt|3\13]] | |||
| | |||
| 438.913 | |||
| | |||
|- | |||
| | |||
| [[7/5]] | |||
| 439.814 | |||
| 1/3-comma | |||
|- | |||
| [[108edt|25\108]] | |||
| | |||
| 440.267 | |||
| | |||
|- | |||
| | |||
| | |||
| 440.340 | |||
| [[Delta-rational chord|DR]] 3:5:7, close to 10/27-comma | |||
|- | |||
| [[95edt|22\95]] | |||
| | |||
| 440.453 | |||
| | |||
|- | |||
| [[82edt|19\82]] | |||
| | |||
| 440.697 | |||
| | |||
|- | |||
| | |||
| [[25/21]] | |||
| 440.760 | |||
| 2/5-comma | |||
|- | |||
| [[69edt|16\69]] | |||
| | |||
| 441.033 | |||
| | |||
|- | |||
| [[56edt|13\56]] | |||
| | |||
| 441.525 | |||
| | |||
|- | |||
| | |||
| [[5/3]] | |||
| 442.179 | |||
| 1/2-comma | |||
|- | |||
| [[43edt|10\43]] | |||
| | |||
| 442.315 | |||
| | |||
|- | |||
| [[73edt|17\73]] | |||
| | |||
| 442.921 | |||
| | |||
|- | |||
| [[30edt|7\30]] | |||
| | |||
| 443.790 | |||
| | |||
|- | |||
| | |||
| 175/81 | |||
| 444.544 | |||
| 2/3-comma | |||
|- | |||
| [[77edt|18\77]] | |||
| | |||
| 444.613 | |||
| | |||
|- | |||
| [[47edt|11\47]] | |||
| | |||
| 445.138 | |||
| | |||
|- | |||
| [[17edt|4\17]] | |||
| | |||
| 447.519 | |||
| | |||
|- | |||
| | |||
| [[35/27]] | |||
| 449.275 | |||
| Full comma | |||
|} | |||
=== Other tunings === | === Other tunings === | ||
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[[Category:Temperaments]] | [[Category:Temperaments]] | ||
[[Category:Tritave-equivalent temperaments]] | [[Category:Tritave-equivalent temperaments]] | ||
Revision as of 08:55, 22 August 2024
Bohlen-Pierce-Stearns (BPS) is a temperament in the 3.5.7 subgroup generated by a sharp ~9/7 (or equivalently a flat ~7/3), tempering out the sensamagic comma, 245/243 so that a stack of two generators represents 5/3 in addition to 81/49, which generates a MOS scale of 4L 5s against the tritave, known as the Bohlen-Pierce Lambda scale. The "canonical" tuning for the generator is 3\13edt, representing the equal-tempered Bohlen-Pierce scale, but a range of other tunings are valid, including 4\17edt, 7\30edt, and 10\43edt.
As the generator of the Bohlen-Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to meantone of the 2.3.5 subgroup, allowing for the tetrad 3:5:7:9 to serve as the theory's primary consonant tetrad.
For technical data, see Sensamagic clan#BPS.
Extensions
Several extensions of this temperament are possible to incorporate additional harmonics.
Interval chains
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 440.6 | 9/7, 35/27 |
| 2 | 881.1 | 5/3, 81/49 |
| 3 | 1321.7 | 15/7, 175/81 |
| 4 | 1762.2 | 25/9, 135/49 |
| 5 | 300.8 | 25/21, 405/343, 875/729 |
| 6 | 741.4 | 75/49, 125/81 |
Tuning spectrum
| Edt Generator |
Eigenmonzo (Unchanged-interval) |
Generator (¢) |
Comments |
|---|---|---|---|
| 5\22 | 432.263 | ||
| 7/3 | 435.084 | 0-comma | |
| 14\61 | 436.514 | ||
| 3645/2401 | 437.449 | 1/6-comma | |
| 23\100 | 437.450 | ||
| 49/45 | 438.632 | 1/4-comma | |
| 3\13 | 438.913 | ||
| 7/5 | 439.814 | 1/3-comma | |
| 25\108 | 440.267 | ||
| 440.340 | DR 3:5:7, close to 10/27-comma | ||
| 22\95 | 440.453 | ||
| 19\82 | 440.697 | ||
| 25/21 | 440.760 | 2/5-comma | |
| 16\69 | 441.033 | ||
| 13\56 | 441.525 | ||
| 5/3 | 442.179 | 1/2-comma | |
| 10\43 | 442.315 | ||
| 17\73 | 442.921 | ||
| 7\30 | 443.790 | ||
| 175/81 | 444.544 | 2/3-comma | |
| 18\77 | 444.613 | ||
| 11\47 | 445.138 | ||
| 4\17 | 447.519 | ||
| 35/27 | 449.275 | Full comma |
Other tunings
- DKW (3.5.7): ~3 = 1\1, ~9/7 = 440.554